In recent years, Noisy Intermediate-Scale Quantum (NISQ) systems are broadly studied, with a particular focus on investigating how near-term devices could outperform classical computers for practical applications. A major roadblock to obtaining a relevant quantum advantage is the inevitable presence of noise in these systems. Therefore, a major focus point of NISQ research is the exploration of noise in currently-available and realistic devices and how the effects of such noise can be mitigated. A growing body of work in this direction proposes various error correcting and error mitigating protocols with an objective to limit this unwanted noise and possibly achieve error suppression. As NISQ devices cannot support full error correction, analysis of the noise and finding ways to suppress it, will increase the chances of obtaining tangible benefits by NISQ computation. In this edition of Active Quantum Research Areas, we cover several recent and promising papers in this direction.

While techniques like Dynamical Decoupling have potential to partially suppress quantum errors, their effectiveness is still limited by errors that occur at unstructured times during a circuit. Furthermore, other commonly encountered noise mechanisms such as cross-talk and imperfectly calibrated control pulses can also decrease circuit execution fidelity. Recent work by [1] discusses an error mitigation strategy named `quantum measurement emulation' (QME), which is a feed-forward control technique for mitigating coherent errors. This technique employs stochastically applied single-qubit gates to ‘emulate’ quantum measurement along the appropriate axis, while simultaneously making this process less sensitive to coherent errors. Moreover, it uses the stabilizer code formalism in order to enable error suppression leading to improved circuit execution fidelity observed in this work. Since QME does not require the computation of correction gates as needed in randomized compiling, it can only protect against errors that rotate the qubit out of the logical codespace. This technique also seems to be effective against coherent errors occurring during twirling gates. For arbitrarily generated circuits, QME can outperform simple dynamical decoupling schemes by addressing discrete coherent errors. Moreover, it does not require costly measurements and feedback is cost-effective as well.

Apart from passively mitigating errors, another approach to rectify these errors will be effective active error-suppressing techniques. Out of the recently introduced methods, virtual distillation is capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Although this technique requires additional (ancilla) qubits, qubit efficiency can be achieved by resetting and reusing qubits. One such method is proposed by [2] named Resource-Efficient Quantum Error Suppression Technique (REQUEST) which is an alternative to virtual distillation methods. For N qubit states, the total qubit requirement of REQUEST is 2N + 1 for any number of copies instead of MN + 1 qubits to use M copies as required by past approaches. The optimal number of these copies is then estimated by using near-Clifford circuits by comparing results mitigated with different values of M to exact quantities. It has been observed that with increasing the optimal number of copies, error suppression will also increase; perhaps exponentially. This suggests that the method can be relevant for larger devices where sufficient qubits and connectivity are available. However, one of the drawbacks of the method is the increase in the overall depth of the quantum circuit in order to achieve a reduction in qubit resources, so further research on this trade-off would be interesting.

Another recent work concerning error suppression is presented in [3]. This work proposes a technique to exponentially suppress bit or phase-flip errors with repetitive error correction. In this work, the authors implement 1D repetition codes embedded in a 2D grid of superconducting qubits. This technique requires the errors to be local and the performance needs to be maintained over many rounds of error correction - two major outstanding experimental challenges. The results demonstrate reduced logical error per round in the repetition code by more than 100× when increasing the number of qubits from 5 to 21. This exponential suppression of bit or phase-flip errors is shown to be stable over 50 rounds of error correction. Also, it was observed that a stable percentage of detection events was observed throughout the 50 rounds of error correction for the system with 21 superconducting qubits, which is important for showing the value of error correction. The authors also perform error detection using a small 2D surface code. Both experimentally implemented 1D and 2D codes agree with numerical simulations considering a simple depolarizing error model, which supports that superconducting qubits may be on a viable path towards fault-tolerant quantum computing. It would be interesting to compare the performance on other types of hardware also.

One of the potential benefits and long-term goals of error correction is attaining scalable quantum computing. However, logical error rates will only decrease with system size while using error correction when physical errors are sufficiently uncorrelated. One limiting factor in terms of scalability is the creation of leakage states, which are non-computational states created due to the excitation of unused high energy levels of the qubits during computation. Particularly for superconducting transmon qubits, this leakage mechanism opens a path to errors that are correlated in space and time. To overcome this, the authors of [4] propose a reset protocol that returns a qubit to the ground state from all relevant higher-level states. It employs a multi-level reset gate using an adiabatic swap operation between the qubit and the readout resonator combined with a fast return. The authors claim a fidelity of over 99% for qubits starting in any of the first three excited states while the gate error is predicted by an intuitive semi-classical model. During the experimentation, only currently existing hardware was used for normal operation and readout. Since there was no involvement of strong microwave drivers which might induce crosstalk, this reset protocol can be implementable on large scale systems. The performance of the protocol is tested with the bit-flip stabilizer code, investigating the accumulation and dynamics of leakage during error correction. The study reveals that applying reset reduces the magnitude of correlations leading to lower rates of logical errors and improved scaling and stability of error suppression as the number of qubits is increased. Therefore, optimizing gates and readout to have minimal leakage is a necessary strategy and the correlated nature of the leakage error makes reset protocols critical for quantum error correction.

Error correction and error mitigation strategies are both valid paradigms which will be required on the road to useful quantum computing. Current NISQ devices, however, cannot support full error correction for deep and wide enough circuits to be useful, therefore more attention has been given to error mitigation strategies that attempt to suppress any type of noise as much as possible. At the moment research is focused in the reduction of noise in gate, initialization and measurement operations, in order to have more reliable information about the state of the qubits during computation. Noise processes like leakage to non-computational states and crosstalk between neighbouring qubits are deemed as extremely important, which led to the proposal of active reset and other qubit control techniques. Experiments with small devices consisting of up to 20 qubits have been performed, in order to: a) show the advantages of error correction in combating leakage and crosstalk in the setting of repeated stabilizer measurements and b) show the advantages of error mitigation through techniques like reset and re-use of qubits and conversion of coherent errors into incoherent. Nevertheless, it is clear that achieving exponential noise suppression in large systems of relevant size is far from straightforward and will require advanced error correction and error mitigation techniques, even though there are indications through experiments with small systems that the aforementioned techniques will provide high-level suppression. As mentioned in [3], there are experimental results on both 1D and 2D codes that show evidence of being within a striking distance of noise suppression (as defined by the surface code threshold).

It will be particularly interesting to see in future research whether a fixed number of physical qubits should best be used for error correction or combined with error mitigation techniques.

References:

[1] Greene et. al., “Error mitigation via stabilizer measurement emulation” arXiv:2102.05767 (Feb. 2021)

[2] P. Czarnik et. al. “Qubit-efficient exponential suppression of errors” arXiv:2102.06056v1 (Feb. 2021)

[3] Z. Chen et. al. “Exponential suppression of bit or phase flip errors with repetitive error correction” arXiv:2102.06132v1 (Feb 2021)

[4] M. McEwen et. al. “Removing leakage-induced correlated errors in superconducting quantum error correction” arXiv:2102.06131v1 (Feb 2021)